The first term on the right hand side of (REF ) simplifies
to \( \log _2 \det \left(2 \pi e \mathbf {v}_{i}\right),\)  where \(\mathbf {v}_{i} \triangleq \mathbb {E}\left\lbrace \alpha _{i}^2\mathbf {x}^{[i]} \mathbf {x}^{[i] {H}}\right\rbrace \)  denotes the
transmit co-variance matrix related to \(\mathbf {x}^{[i]}\)  [1]}. Regarding the equation in (REF ), the second term of the right hand side of (REF ) is upper bounded by the entropy of a Gaussian random variable [2]} as follows
\(&\mathcal {H}(\alpha _{i}\mathbf {x}^{[i]} | y^{[i]}, \hat{\mathbf {h}}) \le \log _2 \det \Big (2 \pi e(\mathbf {v}_{i}-\frac{\mathbf {v}_{i} \hat{\mathbf {h}}^{[i] H} \hat{\mathbf {h}}^{[i]} \mathbf {v}_{i}}{\hat{\mathbf {h}}^{[i]} \mathbf {v}_{i} \hat{\mathbf {h}}^{[i] H}+\Gamma _{i}})\Big ),\) 
